New higher-order implict method for approximating solutions of the initial value problems

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by Taylor’s approach. Specifically, we present an enhanced variant achieved by accelerating the expansion of the Obreschkoff formula. This results in a higher-order implicit corrected method that outperforms Taylor’s method in terms of accuracy. We derive an error bound for the Obreschkoff higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Taylor method. To substantiate our claims, numerical experiments are provided, which highlight the exceptional efficacy of our proposed method over the traditional Taylor method.